"Discrete" means that it can only have certain values, not just any old value. For example, the number of sheep in a field is discrete quantity; you can only have 1 sheep, or 2 sheep, or 3 sheep, etc., but not some fractional number like 1.34995 sheep. Electric charge works the same way: you can only have 1 elementary charge (elementary charge = electron charge), or 2x the elementary charge, or 3x the elementary charge, etc., but not any fractional number.
As for why? Nobody really knows. There is some theoretical speculation of reasons why charge occurs only in integral multiples of the elementary charge, but nothing super-convincing. All we really have to go on is that nobody's ever seen a charge which wasn't an integral multiple of the elementary charge.